According to wikipedia article on polarisation identity, in a normed space , if the parallelogram law holds, then there is an inner product on V such that for all . However, from what I understand, we can prove polarisation identity this way:
However, from this proof, I don't see why we need parallelogram law. On top of that, shouldn't it be the other way round, such that if , then the parallelogram law holds? Can someone explain to me please?